The local discontinuous Galerkin method for convection-diffusion-fractional anti-diffusion equations

In this paper, we consider the discontinuous Galerkin method for solving time dependent partial differential equations with convection-diffusion terms and anti-diffusive fractional operator of order alpha is an element of (1, 2). These equations are motivated by two distinct applications: a dune morphodynamics model and a signal filtering model. The key to study these numerical schemes is to split the anti-diffusive operators into a singular and non-singular integral representations. The problem is then expressed as a system of low order differential equations and a local discontinuous Galerkin method is proposed for these equations. We prove nonlinear stability estimates and optimal order of convergence O(Delta x(k+1/2)) for linear equations and an order of convergence of O(Delta x(k+1/2)) for the nonlinear problem. Finally numerical experiments are given to illustrate qualitative behaviors of solutions for both applications and to confirme our convergence results. (C) 2019 IMACS. Published by Elsevier B.V. All rights reserved.